Someone posted this pronunciation of Latin. They seemed to know what they were talking about. Going by what this says the correct pronunciation would be:
Fay or Fa-dole-lay I guess. Because there is no “ee” sound listed.

Latin Pronunciation (Roman System):

Vowels
Vowel | Short | Long
A: as in pat or father
E: as in pet or they
I: as in fit or machine
O: as in obey (long is the same)
U: as in put or rule

Diphthongs
AE: like ai in aisle
OE: like oi in foil
AU: like ou in out
EI: as in feign
EU: like ew in dew
UI: like we

Consonants
Mostly like in English,
C: always as in cat
G: always as in gun
NGU: as in unguent
QU: as in quit
R: always as in three
J (consonantal I): like y in yet
V (consonantal U): like w in water
BS: like English ps
BT: like English pt
S: always as in this
SU: always as in suave
T: always as in tin
X: always as in extra
Z: like dz in adze

Double consonants are pronounced as one.
There are no translations for articles (a, an, the)

Accent
With words of two syllables, the accent is on the first syllable.
With words of more than two syllables, the accent is on the second to last syllable if the vowel is long, otherwise it is on the second syllable

Turns out it’s not even a 3 syllable word so we’re all wrong. Here’s how a newscaster doing a story on pheidole research pronounced it. I’m assuming she learned how to pronounce it from the researchers? So it’s probably correct? I guess that’s how scallop was pronouncing it as well.